On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation

2016/04/04 by Fashun Gao, Gao, Fashun, Minbo Yang +1 · 47 citations
Computer Science · Mathematics · #35J25 #35J65 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Boundary (topology) #Bounded function #Combinatorics #Domain (mathematical analysis) #Exponent #FOS: Mathematics #Lambda #Lipschitz continuity #Lipschitz domain #Mathematical analysis #Mathematical physics #Mathematics #Nirenberg and Matthaei experiment #Nonlinear Partial Differential Equations #Nonlinear system #Omega #Physics #Quantum mechanics #Sobolev space #Type (biology) #math.AP #msc:35J25 #msc:35J65

paper · pdf · doi:10.48550/arxiv.1604.00826

published in arXiv (Cornell University) (Cornell University) · 30

openalex publication_date 2016/04/04 · arxiv created 2016/06/21 · arxiv updated 2016/06/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation -Δu =(∫_Ω\frac|u|^2_μ^∗|x-y|^μdy)|u|^2_μ^∗-2u+λu\4.14mmin\1.14mm Ω, where Ω is a bounded domain of ℝ^N, with Lipschitz boundary, λ is a real parameter, N≥3, 2_μ^∗=(2N-μ)/(N-2) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

Citations

• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
• Nodal solutions for the Choquard equation
• The Brezis-Nirenberg result for the fractional Laplacian
• A relation between pointwise convergence of functions and convergence of functionals
• On Gravity's role in Quantum State Reduction

Cited by

• Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent
• Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent
• Bifurcation and multiplicity results for critical Grushin-Choquard problems
• Semilinear Elliptic Systems with Nonlocal Homogeneous Critical Nonlinearities
• On nonlinear elliptic problems with Hardy-Littlewood-Sobolev critical exponent and Sobolev-Hardy critical exponent
• Nonlocal problems with Hardy-Littlewood-Sobolev critical exponent and Hardy potential
• Normalized solution for Kirchhoff equation with upper critical exponent and mixed Choquard type nonlinearities
• Infinitely many solutions to a conformally invariant elliptic equation with Choquard-type nonlinearity
• Normalized solutions for fractional Choquard equation with critical growth on bounded domain
• p-biharmonic Kirchhoff equations with critical Choquard nonlinearity
• Quantitative stability of critical points for the nonlocal-Sobolev inequality in Heisenberg group
• Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications
• Normalized solutions for Choquard equations with critical nonlinearities on bounded domains
• Local asymptotics for singular solutions to critical Hartree equations
• Fractional Choquard Equation with Critical Nonlinearities

Related

• Critical exponent Neumann problem with Hardy-Littlewood-Sobolev nonlinearity
• On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents
• Blowing-up solutions for the Choquard type Brezis-Nirenberg problem in dimension three
• Existence of solutions for critical Choquard problem with singular coefficients
• Existence of solutions for critical Choquard equations via the concentration compactness method